In all classical algebraic structures, the Laws of Compositions on a given set are well-defined. But this is a restrictive case, because there are many more situations in science and in any domain of knowledge when a law of composition defined on a set may be only partially-defined (or partially true) and partially-undefined (or partially false), that we call NeutroDefined, or totally undefined (totally false) that we call AntiDefined.

In all classical algebraic structures, the Laws of Compositions on a given set are well-defined. But this is a restrictive case, because there are many more situations in science and in any domain of knowledge when a law of composition defined on a set may be only partially-defined (or partially true) and partially-undefined (or partially false), that we call NeutroDefined, or totally undefined (totally false) that we call AntiDefined.

In this paper we present the development from paradoxism to neutrosophy, which gave birth to neutrosophic set and logic and especially to NeutroAlgebraic Structures (or NeutroAlgebras) and AntiAlgebraic Structures (or AntiAlgebras) that are generalizations and alternatives of the classical algebraic structures.

In this paper we recall, improve, and extend several definitions, properties and applications of our previous 2019 research referred to NeutroAlgebras and AntiAlgebras (also called NeutroAlgebraic Structures and respectively AntiAlgebraic Structures). Let <A> be an item (concept, attribute, idea, proposition, theory, etc.). Through the process of neutrosphication, we split the nonempty space we work on into three regions {two opposite ones corresponding to <A> and <antiA>, and one corresponding to neutral (indeterminate) <neutA> (also denoted <neutroA>) between the opposites}, (...) which may or may not be disjoint – depending on the application, but they are exhaustive (their union equals the whole space). A NeutroAlgebra is an algebra which has at least one NeutroOperation or one NeutroAxiom (axiom that is true for some elements, indeterminate for other elements, and false for the other elements). A Partial Algebra is an algebra that has at least one Partial Operation, and all its Axioms are classical (i.e. axioms true for all elements). Through a theorem we prove that NeutroAlgebra is a generalization of Partial Algebra, and we give examples of NeutroAlgebras that are not Partial Algebras. We also introduce the NeutroFunction (and NeutroOperation). (shrink)

Smarandache in 2019 has generalized the algebraic structures to NeutroAlgebraic structures and AntiAlgebraic structures. In this paper, authors, for the first time, define the NeutroAlgebra of neutrosophic triplets group under usual+ and x, built using {Zn, x}, n a composite number, 5 < n < oo, which are not partial algebras. As idempotents in Zn alone are neutrals that contribute to neutrosophic triplets groups, we analyze them and build NeutroAlgebra of idempotents under usual + and x, which are (...) not partial algebras. We prove in this paper the existence theorem for NeutroAlgebra of neutrosophic triplet groups. This proves the neutrals assocaited with neutrosophic triplet groups in { Zn, X} under product is a NeutroAlgebra of triplets. We also prove the non-existence theorem of NeutroAlgebra for neutrosophic triplets in case of Zn when n = 2p, 3p and 4p (for some primes p). Several open problems are proposed. Further, the NeutroAlgebras of extended neutrosophic triplet groups have been obtained. (shrink)

Neutrosophic theory and its applications have been expanding in all directions at an astonishing rate especially after of the introduction the journal entitled “Neutrosophic Sets and Systems”. New theories, techniques, algorithms have been rapidly developed. One of the most striking trends in the neutrosophic theory is the hybridization of neutrosophic set with other potential sets such as rough set, bipolar set, soft set, hesitant fuzzy set, etc. The different hybrid structures such as rough neutrosophic set, single valued neutrosophic rough set, (...) bipolar neutrosophic set, single valued neutrosophic hesitant fuzzy set, etc. are proposed in the literature in a short period of time. Neutrosophic set has been an important tool in the application of various areas such as data mining, decision making, e-learning, engineering, medicine, social science, and some more. (shrink)

Contemporary architecture is increasingly grounded in science and mathematics. Architectural discourse has shifted radically from the sometimes disorienting Derridean deconstruction, to engaging scientific terms such as fractals, chaos, complexity, nonlinearity, and evolving systems. That's where the architectural action is -- at least for cutting-edge architects and thinkers -- and every practicing architect and student needs to become conversant with these terms and know what they mean. Unfortunately, the vast majority of architecture faculty are unprepared to explain them to students, not (...) having had a scientific education themselves. Here is an architecture book by an architect/scientist, just in time to help architects in the new millennium. Alexander discusses many of the scientific terms arising in cutting-edge architecture, and explains them to those who don't have scientific training or advanced mathematical knowledge. We find discussions of the evolution of forms; the importance of process in design; iteration; genetic algorithms; sequences of transformations; different levels of scale (i.e. fractals); etc. They are explained here by an architect who is also a scientist, because he wants to change the way architects think and build. Alexander is not merely popularizing other scientists' results and making them accessible to architects: he is in fact presenting new and original scientific work that ties many of these concepts together in a way that will be useful to architects. (shrink)

The article presents proofs of the context freeness of a family of typelogical grammars, namely all grammars that are based on a uni- ormultimodal logic of pure residuation, possibly enriched with thestructural rules of Permutation and Expansion for binary modes.

Neutrosophy, the study of neutralities, is a new branch of Philosophy that has applications in many different fields of science. Inspired by the idea of Neutrosophy, Smarandache introduced NeutroAlgebraicStructures (or NeutroAlgebras) by allowing the partiality and indeterminacy to be included in the structures’ operations and/or axioms.

In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, and the (...) NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world. (shrink)

In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry.

In this paper we extend Neutro-Algebra and Anti-Algebra to geometric spaces, founding Neutro/Geometry and AntiGeometry. While Non-Euclidean Geometries resulted from the total negation of a specific axiom (Euclid's Fifth Postulate), AntiGeometry results from the total negation of any axiom or even more axioms of any geometric axiomatic system (Euclidean, Hilbert, etc. ) and of any type of geometry such as Geometry (Euclidean, Projective, Finite, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.), and Neutro-Geometry results from the partial negation of one (...) or more axioms [and without total negation of any axiom] of any geometric axiomatic system and of any type of geometry. Generally, instead of a classical geometric Axiom, one can take any classical geometric Theorem of any axiomatic system and of any type of geometry, and transform it by Neutrosophication or Antisofication into a Neutro-Theorem or Anti-Theorem respectively to construct a Neutro-Geometry or Anti-Geometry. Therefore, Neutro-Geometry and Anti-Geometry are respectively alternatives and generalizations of Non-Euclidean Geometries. In the second part, the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra and Anti-Algebra, then to Neutro-Geometry and Anti-Geometry, and in general to Neutro-Structure and Anti-Structure that arise naturally in any field of knowledge is recalled. At the end, applications of many Neutro-Structures in our real world are presented. (shrink)

The paper has three parts. In the first part ExtOSR, an extended version of Ontic Structural Realism, will be introduced. ExtOSR considers structural properties as ontological primitives, where structural properties are understood as comprising both relational and structurally derived intrinsic properties or structure invariants. It is argued that ExtOSR is best suited to accommodate gauge symmetry invariants and zero value properties. In the second part, ExtOSR will be given a Humean shape by considering structures as categorical and global. It (...) will be laid out how such structures serve to reconstruct non-essential structural kinds and laws. In the third part Humean structural realism will be defended against the threat of quidditism. (shrink)

The ‘ontic’ form of structural realism , roughly speaking, admits a complete elimination of the objects in the discourse of scientific theories, leaving us with structures only. As put by the defenders of such a claim, the idea is that all there is are structures and, if the relevant structures are to be set-theoretical constructs , as it has also been claimed, then the relations which appear in such structures should be taken to be ‘relations without the relata’. As far (...) as we know, there is not a definition of structure in standard mathematics which fits their intuitions, and even category theory seems not to correspond adequately the OSR claims. Since OSR is also linked with the semantic approach to theories, the structures to be dealt with are taken to be set-theoretical constructs. But these are ‘relational’ structures where the involved relations are built from basic objects , and so no complete elimination of the relata is possible, although it would be adequate for characterizing OSR. In this paper we present a definition of a kind of relation that does not depend on the particular objects being related in the sense that the ‘relation’ continues to hold even if the relata are exchanged by suitable objects. Although there is not a ‘complete’ elimination of relata, our definition might be viewed as an alternative way of finding adequate mathematical ‘set-theoretical’ frameworks for describing at least some of the intuitions regarding OSR within a ‘set-theoretical’ framework. (shrink)

Structural realism has rapidly gained in popularity in recent years, but it has splintered into many distinct denominations, often underpinned by diverse motivations. There is, no monolithic position known as ‘structural realism,’ but there is a general convergence on the idea that a central role is to be played by relational aspects over object-based aspects of ontology. What becomes of causality in a world without fundamental objects? In this book, the foremost authorities on structural realism attempt to answer this and (...) related questions: ‘what is structure?’ and ‘what is an object?’ Also featured are the most recent advances in structural realism, including the intersection of mathematical structuralism and structural realism, and the latest treatments of laws and modality in the context of structural realism. The book will be of interest to philosophers of science, philosophers of physics, metaphysicians, and those interested in foundational aspects of science. (shrink)

The Russian Genitive of Negation construction (Gen Neg) involves case alternation between Genitive and the two structural cases, Nominative and Accusative.1 The factors governing the alternation have been a matter of debate for many decades, and there is a huge literature. Here we focus on one central issue and its theoretical ramifications. The theoretical issue is the following. The same truth-conditional content can often be structured in more than one way; we believe that there is a distinction between choices in (...) how to structure a situation to be described, and choices in how to structure a sentence describing the (already structured) situation. The distinction may not always be sharp, and the term Information Structure may perhaps cover both, but we believe that the distinction is important and needs closer attention. Babby (1980), in a masterful work on the Russian Genitive of Negation, argued that the choice depended principally on Theme-Rheme structure; after initially following Babby (Borschev & Partee 1998), we later argued (Borschev & Partee 2002a,b) that the choice reflects not Theme-Rheme structure but a structuring of the described situation which we call Perspectival Structure. Here we briefly review the phenomenon, Babby’s Theme-Rheme-based analysis, and our arguments for a different analysis. We then consider Hanging Topics, partitive Genitives, and broader licensing conditions of Genitive case, raising the possibility that our counterexamples to Babby’s use of Theme-Rheme structure might be explained away as examples involving Hanging Topics rather than (Praguian) Themes. We argue against that idea as well, but leave open the possibility that our Perspectival Structure may eventually be construable as a kind of information structure itself, if that notion can include some kinds of structuring of the situation as well as of the discourse. (shrink)

Some illnesses, e.g. OCD and schizophrenia, corrupt the activity mediated by one's basic psychological framework but do not corrupt that framework itself. But some illnesses, e.g. psychopathy, corrupt that framework itself. Thus, whereas OCD and schizophrenia are structure-internal mental illnesses, psychopathy is a structural mental illness. Structure-internal mental illnesses can be alleviated, but structural mental illnesses cannot.

We study first-order expansions of ordered fields that are definably complete, and moreover either are locally o-minimal, or have a locally o-minimal open core. We give a characterisation of structures with locally o-minimal open core, and we show that dense elementary pairs of locally o-minimal structures have locally o-minimal open core.

If categorical equivalence is a good criterion of theoretical equivalence, then it would seem that if some class of mathematical structures is represented as a category, then any other class of structures categorically equivalent to it will have the same representational capacities. Hudetz (2019a) has presented an apparent counterexample to this claim; in this note, I argue that the counterexample fails.

In 1982, Wolniewicz proposed a formal ontology of situations based on the lattice of elementary situations (cf. [7, 8]). In [3], I constructed some types of formal structure Porphyrian Tree Structures (PTS), Concepts Structures (CS) and the Structures of Individuals (U) that formally represent ontologically fundamental categories: species and genera (PTS), concepts (CS) and individual beings (U) (cf. [3, 4]). From an ontological perspective, situations and concepts belong to different categories. But, unexpectedly, as I shall show, some variants of (...) CS and Wolniewicz’s lattice are similar. The main theorem states that a subset of a modified concepts structure (called CS+) based on CS fulfils the axioms of Wolniewicz’ lattice. Finally, I shall draw some philosophical conclusions and state some formal facts. (shrink)

Kuhn’s Structure of Scientific Revolutions has been attacked for many reasons. Key analytic terms, most importantly “paradigm,” were widely regarded as poorly defined. To many readers Structure seemed to suggest that the process of theory change is irrational, or at least non-rational. And even his characterization of normal science seemed to some readers to paint a very unflattering picture of scientists as excessively dogmatic and uncritical. More recently, Lorraine Daston has argued that the notion of “structure” that (...) figures in the title of the book, as well as in the analysis of the history of science discussed throughout the book, is “dusty and dated”. Historians of science tend to focus on the particular, in recognition that the sorts of events they study are unique. My aim in this paper is to defend Kuhn’s appeal to the notion of structure. First, I argue that Structure of Scientific Revolutions is not primarily a contribution to the history of science, despite the fact that it cites many historical sources and articles in the history of science. Consequently, it is a mistake to regard Kuhn’s appeal to the notion of structure as a case of badly written history of science. Second, I argue that insofar as the book is a contribution to philosophy, the sort of thing Kuhn means by “structure” is perfectly respectable, and is often presupposed in many philosophical studies, especially in general philosophy of science. Third, I defend Kuhn’s analysis of science, especially the enterprise of identifying the structure of scientific revolutions. (shrink)

The problem of structure and agency has been the subject of intense debate in the social sciences for over 100 years. This book offers a solution. Using a critical realist version of the theory of emergence, Dave Elder-Vass argues that, instead of ascribing causal significance to an abstract notion of social structure or a monolithic concept of society, we must recognise that it is specific groups of people that have social structural power. Some of these groups are entities (...) with emergent causal powers, distinct from those of human individuals. Yet these powers also depend on the contributions of human individuals, and this book examines the mechanisms through which interactions between human individuals generate the causal powers of some types of social structures. The Causal Power of Social Structures makes particularly important contributions to the theory of human agency and to our understanding of normative institutions. (shrink)

At the time of his death in May 1961, Maurice Merleau-Ponty held the chair of Philosophy at the College de France. Together with Jean-Paul Sartre and Simone de Beauvoir, he was cofounder of the successful and influential review Les Temps Modernes. However, after Merleau-Ponty's two studies of Marxist theory and practice (Humanisme et Terreur and Les Aventures de la Dialectique), he alienated both orthodox Marxists and "mandarins of the left" such as Sartre and de Beauvoir. Perhaps his most lasting contribution (...) to the interpretation of human existence was his formulation of a positive philosophical approach to psychology the foundations of which were clearly outlined in The Structure of Behavior and later more fully developed in Phenomenology of Perception. (shrink)

In this paper, I argue in favor of a structural turn in social contract theory. More precisely, I argue that dealing with the complex and dynamic nature of the social world requires an emphasis on social structures greater than what contractarians often consider. I take structures to be the dynamic and non-random networks of interdependence among all active components that shape society. I also constrain my focus to a growing body of literature on diversity that explores plausible contractarian alternatives given (...) the complexity of modern societies. I argue that shifting weight from diversity to structures in theorizing alternative contractarian frameworks would be an improvement. (shrink)

Whether or not quantum physics can account for molecular structure is a matter of considerable controversy. Three of the problems raised in this regard are the problems of molecular structure. We argue that these problems are just special cases of the measurement problem of quantum mechanics: insofar as the measurement problem is solved, the problems of molecular structure are resolved as well. In addition, we explore one consequence of our argument: that claims about the reduction or emergence (...) of molecular structure cannot be settled independently of the choice of a particular resolution to the measurement problem. Specifically, we consider how three standard putative solutions to the measurement problem inform our under- standing of a molecule in isolation, as well as of chemistry’s relation to quantum physics. (shrink)

The central problem of social theory is 'structure and agency'. How do the objective features of society influence human agents? Determinism is not the answer, nor is conditioning as currently conceptualised. It accentuates the way structure and culture shape the social context in which individuals operate, but it neglects our personal capacity to define what we care about most and to establish a modus vivendi expressive of our concerns. Through inner dialogue, 'the internal conversation', individuals reflect upon their (...) social situation in the light of current concerns and projects. On the basis of a series of unique, in-depth interviews, Archer identifies three distinctive forms of internal conversation. These govern agents' responses to social conditioning, their individual patterns of social mobility and whether or not they contribute to social stability or change. Thus the internal conversation is seen as being the missing link between society and the individual, structure and agency. (shrink)

It is often argued that our obligations to address structural injustice are collective in character. But what exactly does it mean for ‘ordinary citizens’ to have collective obligations visà- vis large-scale injustice? In this paper, I propose to pay closer attention to the different kinds of collective action needed in addressing some of these structural injustices and the extent to which these are available to large, unorganised groups of people. I argue that large, dispersed and unorganised groups of people are (...) often in a position to perform distributive collective actions. As such, ordinary citizens can have massively shared obligations to address structural injustice through distributive action, but, ultimately, such obligations are ‘collective’ only in a fairly weak sense. (shrink)

The epistemic probability of A given B is the degree to which B evidentially supports A, or makes A plausible. This paper is a first step in answering the question of what determines the values of epistemic probabilities. I break this question into two parts: the structural question and the substantive question. Just as an object’s weight is determined by its mass and gravitational acceleration, some probabilities are determined by other, more basic ones. The structural question asks what probabilities are (...) not determined in this way—these are the basic probabilities which determine values for all other probabilities. The substantive question asks how the values of these basic probabilities are determined. I defend an answer to the structural question on which basic probabilities are the probabilities of atomic propositions conditional on potential direct explanations. I defend this against the view, implicit in orthodox mathematical treatments of probability, that basic probabilities are the unconditional probabilities of complete worlds. I then apply my answer to the structural question to clear up common confusions in expositions of Bayesianism and shed light on the “problem of the priors.”. (shrink)

What is structural rationality? Specifically, what is the distinctive feature of structural requirements of rationality? Some philosophers have argued, roughly, that the distinctive feature of structural requirements is coherence. But what does coherence mean, exactly? Or, at least, what do structuralists about rationality have in mind when they claim that structural rationality is coherence? This issue matters for making progress in various active debates concerning rationality. In this paper, I analyze three strategies for figuring out what coherence means in the (...) debates on structural rationality. I argue that these strategies face problems. (shrink)

Structural realism is considered by many realists and antirealists alike as the most defensible form of scientific realism. There are now many forms of structural realism and an extensive literature about them. There are interesting connections with debates in metaphysics, philosophy of physics and philosophy of mathematics. This entry is intended to be a comprehensive survey of the field.

Although they are continually compositionally reconstituted and reconfigured, organisms nonetheless persist as ontologically unified beings over time – but in virtue of what? A common answer is: in virtue of their continued possession of the capacity for morphological invariance which persists through, and in spite of, their mereological alteration. While we acknowledge that organisms‟ capacity for the “stability of form” – homeostasis - is an important aspect of their diachronic unity, we argue that this capacity is derived from, and grounded (...) in a more primitive one – namely, the homeodynamic capacity for the “specified variation of form”. In introducing a novel type of causal power – a „structural power‟ – we claim that it is the persistence of their dynamic potential to produce a specified series of structurally adaptive morphologies which grounds organisms‟ privileged status as metaphysically “one over many” over time. (shrink)

Current approaches used in educational research and practice to evaluate the quality of written arguments often rely on structural analysis. In such assessments, credit is awarded for the presence of structural elements of an argument, such as claims, evidence, and rebuttals. In this article, we discuss limitations of such approaches, including the absence of criteria for evaluating the quality of the argument elements. We then present an alternative framework, based on the Rational Force Model (RFM), which originated from the work (...) of a Nordic philosopher Næss. Using an example of an argumentative essay, we demonstrate the potential of the RFM to improve argument analysis by focusing on the acceptability and relevance of argument elements, two criteria widely considered to be fundamental markers of argument strength. We outline possibilities and challenges with using the RFM in educational contexts and conclude by proposing directions for future research. (shrink)

Why do speakers of all languages use different grammatical structures under different communicative circumstances to express the same idea? In this comprehensive study, Professor Lambrecht explores the relationship between the structure of sentences and the linguistic and extra-linguistic contexts in which they are used. His analysis is based on the observation that the structure of a sentence reflects a speaker's assumptions about the hearer's state of knowledge and consciousness at the time of the utterance. This relationship between speaker (...) assumptions and formal sentence structure is governed by rules and conventions of grammar, in a component called 'information structure'. Four independent but interrelated categories are analysed: presupposition and assertion, identifiability and activation, topic, and focus. (shrink)